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FULL DISCLOSURE

If you click through an Amazon link on this blog and subsequently make a purchase, I will receive a small credit from Amazon. I will be very grateful for this credit and will use it purchase still more books and such to share with you. An eternal circle of Amazon life, you might say:-)

Gnomes and Gnumbers: A Mathematical Tale

Once upon a time, in the deepest, darkest recesses of the earth, there lived a clan of four greedy gnomes. The first was as blue as a crisp, clear sky. The second was the shade of a red apple in autumn. The third was as yellow as the sunand the fourth was greener than a blade of grass.

Now gnomes are a curious folk. Burrowing deep in the ground,
they build hidden cities that guard the treasures of the earth. Of
course you already knew that, but these gnomes were particularly curious
folk. Not content to work with their fellow earth-dwellers, they
wanted to make a name for themselves and build up a treasure greater
than any the world had ever known. So these four miserly gnomes dug
and burrowed and tunneled until they reached the middle of the earth
where they believed their treasure would remain protected.

Days and weeks and months went by and the four gnomes came to
realize that they had nothing to do but count and sort their mounds and
mounds of jewels all the day long. It wasn't long before the gnomes
began arguing among themselves about the true owner of the treasure.

"It is all mine!" squeaked the small yellow gnome as he scooped
great piles of rainbow-colored gems toward himself. "I am the one who
found the way to the middle of the earth. The jewels belong to me."

After many days of bickering and quarreling, each of the four gnomes
had clearly argued their reason for being the rightful owner of the
treasure. Finally, the red gnome offered a solution.

"Perhaps if we divide the gems evenly among ourselves..." he
timidly suggested. (Having kept some of his wits about him, he realized
that the only way for peace and quiet to return to their cavernous home
was to share the treasure equally.)

"But how shall we know if we have divided the treasure evenly?" asked the green gnome.

The gnomes fell silent, each trying to think of a way to be
absolutely certain that their pile was equal to the piles of the other
gnomes.

After what seemed like hours upon hours, the yellow gnome
suggested that they take a trip back up through the winding tunnels to
the surface of the earth in search of someone with wisdom who could
solve their problem. The others reluctantly agreed and soon they were
on their way.

The gnomes made their way through a tunnel which led to a secret
hollow in a magnificent, ancient tree. Climbing out of the hollow,
they tripped over the gnarled feet of the tree and looked around hoping
to find someone to direct them to the dwelling place of a wise one who
would offer a solution to their problem. Suddenly a sparkle of
brilliant color caught the eye of the green gnome.

"What is that?" he asked in a whisper.

The gnomes turned and together discovered a pile of the most beautiful gemstones.

"They are mine!" shouted the green gnome. "I spotted them first!"

Before he had a chance to dive upon the pile of gems, the red
gnome held up his wee hand and interrupted. "Let us use sticks to
count how many gems there are and then we shall share them. Surely this
is a great treasure which we must carefully guard." Then reaching up
to the tree above him, he pulled down on a branch and snapped it off.

"Huuuummmphh!"

"What was that?" asked the blue gnome.

The other gnomes ignored the deep and unusual sound and went about their work of counting.

After spreading the gems on the ground, the gnomes laid down the
branch next to the first stone. And when they added a second gem they
broke the branch in two. For the third stone they broke one of the two
branches again and saw that they had three stones and three sticks.
Then a fourth gem was moved but the sticks were becoming hard to keep
track of and they stopped to find a solution.

"Let us use two of the pieces of the branch to make the shape of
the Letter V. That looks like the "V" between our thumb and other
fingers when we hold up our hand. That will help us to remember the
number five" suggested one of the gnomes.

Placing one of the leftover sticks before the V-shaped pair, the gnomes counted four.

"One before five is four."

The gnomes continued to count the gems in this way. The
V-shaped formation showed five and then adding a stick after it made
six.

"One after five is six." The gnomes were rather pleased with themselves.

Pulling a seventh gem from the group, they decided to break the
single stick in half to make seven. One of those was broken again to
make eight.

"This is getting confusing again," sighed the red gnome. "What can we do to count so many gemstones?

"I know," boasted the greedy yellow gnome. "Let's cross two
sticks together, like our arms crossed in front of our chest. This
will remind us that the gems are all mine!"

The other three were not happy with his solution, but not being
able to come up with a better one, they agreed. It was true, when both
hands are crossed together they counted ten fingers. The sticks in the
shape of a V were rearranged to form an X.

"One before ten is nine" said the yellow gnome.

Again the gnomes continued moving the gems and counting. The
X-shaped sticks by themselves represented ten gems and then one more
was added in front of it.

"One after ten is eleven. And two after ten is twelve."

The gnomes were tired and sat down to rest.

"We have so many more gems to count. We will most definitely need another branch" announced the yellow gnome.

And just as he raised his arm to grab the nearest
branch..."Huuuummmphh!" An enormous woody hand swooped down and
scooped him up high above the green ceiling of the forest.

LESSON PLANS

According to the age and ability of your child, this lesson
could be used over a period of one week or up to one month. All work
should be done in a Main Lesson Book with both stick and block
crayons. Recommended resources and suggested sources are linked below.The
lesson plans are suggested for two different age groups. The ages are
only offered as a general estimate but, of course, the parent will be
able to best identify the work which will most benefit her own child.

The original art work can be downloaded and reprinted for use as a visual in telling this story.

If you are using this math story with a 5 or 6 year old
(or a child with little or no previous math experience), you may want
to consider introducing one Roman Numeral per day. Before you tell
this story, go outside with your child and find a stick to use for
story time. The child can break it apart as the gnomes break their
stick. (Rather than breaking a stick, using toothpicks is another
option.) The illustration offered above of the gemstones, sticks and
Roman Numerals is a sample page of a Main Lesson Book. For a young
child you will want to demonstrate drawing this page and allow your
child to imitate you.

After the Roman Numerals I through XII are introduced, you will want
to spend a few days writing the numbers 1 through 12 on another page of
the Main Lesson Book. Ask your child to draw a different colored
gemstone between each number to keep them from running together into
one huge number.

As you work through this introduction you should spend some part of
your lesson playing counting games. Find creative ways of reinforcing
number recognition. Model numbers one through twelve out of modeling
beeswax or instead, cut them in salt dough with number cookie cutters,
which can be baked and later painted.

Add movement to your child's learning and take time to play number
games. Toss a bean bag back and forth while taking turns counting the
next number. (Say "One" and toss the bean bag to your child. He or
she will respond "Two" and throw it back to you as you say "Three" and
so on.) The early learning game of Walking the Line can offer another
fun exercise in counting. Mark a line with masking tape or use a grout
line on a tile flour. As your child walks forward, together count
forward to twelve and as he or she walks backwards, count the numbers
backwards with your child from twelve to one.

If you are using this math story with a 7 or 8 year old
(or a child who has a basic understanding of numbers and familiarity
with one or two of the four basic math processes), you may be able to
tell this story in one day. Have your child complete the entire Main
Lesson Book entry of Roman Numerals I through XII on the same or
following day. (As with a younger child, some children will need to
imitate your example. Don't hesitate to draw with your child.) On the
following day ask your child to narrate the story as you look at
yesterday's drawing in the Main Lesson Book.

According to the ability of your child, have him write (in crayon)
the numbers one through one hundred in his Main Lesson Book. (You can
spread this exercise out of several days if necessary.) Ask him to
change colors every tenth number. (For example, 1 through 9 are green,
10-19 are red, 20-29 are blue, etc...) This will help in preparation
for learning place value. Have him separate each number by drawing a
gemstone between each number to keep the numbers from running
together.

One day, or if necessary over the course of several days, have your
child model the numbers one through twelve using beeswax or use cookie
cutters and salt dough and bake.

SUGGESTED RESOURCES AND RECOMMENDED SUPPLIERS

Buy the best art supplies that you are able to afford. The
following are suggestions and substitutions should be made according to
the preferences and resources of each individual family.

Main Lesson Book

Traditional Main Lesson Books with onions skins separating the pages
(to resist crayon smudges between pages) are available from A Toy Garden.
(You may want to consider buying two if you choose this option.)
Another suggestion is to use an artist's sketch book or simply collect
pages in a binder or album, separated by plastic page protectors.

Beeswax Crayons

Nothing compares to beeswax crayons. The smell of sweet honey and
the glide of application make using them an enjoyable sensory
experience. Unlike Crayola and other popular brands of crayons, the
color of beeswax crayons can be blended on the page. The pages in the
Main Lesson Book should be drawn with stick crayons and the backgrounds
should be shaded with block crayons. Encourage your child to never
leave white on her page. Each entry is a work of art and should be
approached as so. These art supplies are special and the child should
be encouraged to treat them with care. We recommend either Stone or
Stockmar which is available from Paper, Scissors, Stone.

Handcraft Supplies

Spend time with your child in making his own set of math gnomes.
Use wooden bases and natural wool felt. Again, the natural fibers are
a sensory experience that enrich the child's learning and train the
senses. Handcraft supplies are available from A Child's Dream Come
True (Scroll to the bottom of this page for the wooden dwarf bodies and this page
for plant-dyed, wool felt. If these supplies exceed your budget, the
pictures can serve as a help in finding less expensive alternatives at
your local craft store.)

Blackboard and Pastels

Using artist's pastels and a quality blackboard allows the written
experience of math to tend toward the beautiful and creative. Unlike
ordinary colored chalk, artist's pastels glide on the surface.
Excellent quality blackboards and pastels are available from Paper, Scissors, Stone.

Modeling Beeswax

Modeling beeswax is a treat for the senses. From the sweet
fragrance of honey to its response to the warmth of the hands, this
medium adds much to the learning experience. We recommend Stockmar Modeling Beeswax which is available from Paper, Scissors, Stone.

Children learn abstract concepts best when they are able to
visualize them and use hands-on manipulatives. At your local craft
store you can purchase bags of glass gemstones.
You will need at least one hundred stones for this year's math work.
Be sure to keep this choking hazard away from small children.

Sorting Baskets

The gnomes will need baskets to sort their gems. Magic Cabin sells miniature baskets
as part of their tree fort collection. You will need twenty baskets
(two sets from Magic Cabin) to work on place value work with the
gemstones.

Tube Watercolors,Watercolor Brushes and Paper, and a Watercolor Board

Tube watercolors are a pleasure to work with and quality paper and
brushes make a huge difference. Using a watercolor board offers a
smooth surface on which to work and a convenient spot for a work of art
to dry. These boards are also easily wiped clean and protect kitchen
tables from stains.

Paper, Scissors, Stone sells Stone watercolors
which are of excellent quality. Simply squeeze a small amount (the
size of a dime) in the bottom of a small mason jar and fill with an
inch of water. Don't mix the paint. Use the tinted water for a
lighter color effect and lightly touch the brush to the small mound of
paint ("the worm") at the bottom of the jar for a deeper hue.

Small Bean Bags

Ordinary bean bags can be used for number tossing games. Counting,
skip counting and math facts can be practiced while having fun with
this movement. You can splurge with an herbal-filled set or enjoy a handcraft project with your child and knit your own using the simple patterns in Melanie Falick's Kids Knitting. Perhaps you can come up with a creative way to glue, sew or knit the different math process symbols on different bean bags.

Index Cards

Last but not least, just an ordinary set of unlined index cards available at any office supply store.

Be sure to check back regularly as the Gnome and Gnumbers Album on the left sidebar is updated with Main Lesson Book samples.

The yellow gnome held his eyes closed tightly as the woody hand
lifted him up, high above the treetops. Suddenly everything was still
and quiet except for the steady sound of a groaning wind that was
blowing his pointed hat directly behind his head. Mustering up enough
courage to open just one eye, the gnome saw before him a tree that
looked like a man. Or was it a man who looked like a tree? As these
questions danced in his head, the Tree Man spoke,

“You have taken that which does not belong to you” he groaned in his deep, woodsy voice.

“I…I’m...I’m sure I don’t know what you mean, thank you,” replied the gnome, hoping that being polite would help matters.

The gnome was completely confused by the Tree Man’s strange
words and, wondering what had happened to his three friends far below,
he ignored the woeful words of the Tree Man and begged to be returned
to the ground.

“Please, put me down on the ground with my friends. I am a
gnome, an earth-dweller, and it is not natural for me to be suspended
at such great heights.”

With that the gnome heard the familiar groan that had sounded when the branch had been snapped from the tree.

“Huuuummmphh! And neither is it natural for my ancient branches to be broken into bits” replied the Tree Man.

With these words he lowered his enormous wooden hand and soon the
gnome was safely back on the ground with his three companions.
Suddenly the Tree Man took a step forward into a clearing and the four
gnomes could plainly see his enormous stature.

Before they had a chance to speak, the Tree Man addressed them,

“I am Old Dismas. I have lived in these woods since it came to
be. When the earth was still young, a wizard cast an enchantment on my
wood,

With those words, Old Dismas left the gnomes in the clearing. The
earth groaned under the weight of his steps and the quaking sound of
the forest floor echoed throughout the woods.

"What are we to do? Oh dear! Oh me! We've brought a terrible enchantment upon ourselves!" cried the blue gnome.

"It sounds as if we will have to find something to count. The
Ancient Tree said that the enchantment will not break until we discover
the secret of numbers. What shall we count?"

The gnomes confusedly looked around until they again spotted the mountain of gemstones.

"The gemstones! That's what we shall count" shouted the red gnome, with much excitement.

The four gnomes approached the huge mountain of gemstones and
noticed that there were small baskets scattered around the ground.
Each of them grabbed one and collected as many gems as he could fit in
his basket.

After their baskets were filled to the brim, the green gnome asked, "Now what do we do?"

"How about we break a branch off that tree over there and use
the sticks to count our gemstones" replied the blue gnome, with a look
of enthusiasm.

"No!" shouted the other three. "Don't you remember? That's what got us in this mess in the first place!"

The four gnomes didn't know what to do, so they emptied their
baskets of gemstones and began spreading them on the ground in order to
count. The blue and green gnomes decided to sort their gemstones into
pairs of two, but soon the blue gnome became frustrated.

"Well now, this is rather ODD. Every time I try to sort my gems in pairs, I always find one left over."

The green gnome continued his work of sorting and soon matched
each gemstone perfectly in pairs. Upon seeing his success, the blue
gnome grumbled,

"Now, that is not EVEN fair. You do not have any left over. How
is it that when I match my gemstones in pairs I always have one left
over and you are able to match every last one?"

Soon the crunching and snapping of twigs could be heard again
beneath the feet of Old Dismas. As he was approaching the clearing, he
overheard the debate and offered an explanation.

"Yes, it is ODD when a group has one left over and for this
reason we call those numbers ODD. How many gemstones have you counted?"
asked Old Dismas.

"Eleven" replied the blue gnome. "Is eleven an ODD number?"

"Yes" answered the Ancient Tree. "And how many gemstones have you counted?" he then asked the green gnome.

"Twelve. Twelve gemstones" answered the gnome. "Twelve is an
EVEN number then because all of my gems have a partner. That's it!
I've discovered the secret of numbers!" The silly green gnome began
dancing around his sorted gemstones.

"Not so fast," laughed Old Dismas. "The secret is far deeper
than the discovery of ODD and EVEN. But don't loose hope. There is
someone here to help you."

And suddenly a King dressed in purple was standing before them.

To Be Continued...

LESSON PLANS

This lesson continues the story from Lesson 1 and introduces the
concept of odd and even numbers. If your child has not yet mastered
the material presented in Lesson 1, wait before continuing with Lesson
2. Regardless, remember to practice and review Roman Numerals
throughout the week. If your child is interested, introduce further Roman Numerals
such as L equals 50 and C equals 100. Use your child's work in the
Main Lesson Book as a springboard for review. By creating a Main
Lesson Book, your child is writing and illustrating his own Math book!

If
you are continuing with a 5 to 6 year old, rely heavily on the
manipulatives (the gemstones) to teach this concept. Talk to your
child about the things around her that naturally exist in pairs: two
eyes, two ears, two feet, two socks or shoes. Then proceed to group
the gemstones in pairs of two. Fill up the miniature baskets with
gemstones (keep it under 10 for a young child) and ask him to sort them
in pairs. If one is left over you can make a game of saying, like the
blue gnome, "Well, isn't this ODD!" Point out that when all the
gemstones have a partner, they are all EVEN. Make an Odd Number and
Even Number page in your child's Main Lesson Book modeled after the
sample page in the story. Draw with your child if he is having trouble
expressing these concepts in writing and allow your child to spend a
few days on his page if necessary. Young children learn best by
imitation.

If you're working with a 7 to 8 year old, repeat the above activity
and review it each day. Create and add the Even and Odd Number pages
in the Main Lesson Book. Encourage your child to decorate his page with
a border that reflects the story or concept learned. Remember, these
pages are to be approached as works of art. When your child has grasped
the concepts of even and odd, continue the work by using the
blackboard. Write a random number and ask your child to tell you if it
is even or odd. Allow her to use the gemstones to find the answer.

Continue to combine movement with your child's learning. Games of
bean bag toss are a fun way to practice numbers. This week, play games
of Even and Odd toss. As you toss the bean bag to your child say,
"One!" and as your child catches it, he responds, "Three!" Continue
through the odd numbers as he is able and then try the even numbers.
Weather permitting, take your lessons outside. Use stones or acorns and sidewalk chalk to make number pictures on the ground.

Beginning this week we are going to concentrate on a different number
each week. Don't skip this activity, even if your child knows his
numbers well and can work basic math equations. We're going to be
considering the concrete reality behind the very abstract idea of
numbers. This week we'll begin with Number 1.

Consider the world and its Creator. What exists that expresses oneness?
Some answers could be one God, one sun, one moon, one nose, etc...
(Remember these answers are taken from the experience of a young
child. There is no need to correct him and point out that the sun is a
star and there are many in the universe, etc...) Ask your child to
make a Number 1 page in his Main Lesson Book.

GNOMES AND GNUMBERS STORYBOOK

It is highly encouraged that you print out these math stories and
artwork and read-aloud to your child in a comfortable setting and for
this reason we are offering them in PDF format for free download. For
the story files, the larger font size was chosen so that the child (or
an older sibling) can go back and read the stories on his own. Keep
them in a 3-ring binder, each page stored in it's own page protector so
the child can flip through without worrying about ripping pages.
Include the artwork throughout the story and soon your child will have
a storybook of his own.

You can make this storybook as simple or detailed as you wish. The
cover page shown here can be printed for use in a binder or perhaps you
will want to get creative and make a felt relief on an album using
ideas from a book such as Felt Wee Folk.

About the name Old Dismas: This name was chosen in honor of the repentant thief on the Cross, St. Dismas.
The character in this tale not only exhibits wisdom to those who
struggle with the passion of greed, but offers it with patience and
love. For this reason it seemed appropriate for the wise Tree Man of
our story to share a name with the grace-filled man who from a tree
offered us the timeless example, "Lord, remember me when you come into
your Kingdom" (Luke 23:42).

Like the thief will I
confess thee: Remember me, O Lord, in thy Kingdom.-Prayer Before Communion, St. John Chrysostom

By this time the clearing was littered with baskets thrown here and
there. Some were filled with gemstones and others had been toppled
over. The mountain of gemstones was still high above the gnomes, but
gems were now scattered everywhere, some in even numbered piles and
some mismatched in odd groupings. The gnomes knew that they must
continue counting in order to find the secret of numbers, but they were
getting more and more frustrated the longer they tried to count. And
while they were happy that Old Dismas had explained the difference
between odd and even numbers, they still knew that there were far more
gemstones than they were able to count on their own.

The gnomes were surprised when they suddenly saw a purple-clad
king come out from behind the great trunk legs of Old Dismas.

"There is someone here for you to meet," the Ancient Tree announced to the gnomes.

The king stepped out in the clearing. He was an odd looking
fellow, almost as wide as he was tall. He was dressed in purple robes
that seemed to reflect every color imaginable and, instead of a crown,
he wore a pointed cap trimmed in gold.

Careful not to trip over the gemstones and baskets, he began,

"I am King Equals, King of all Numeria. It was my ancestor who
cast the enchantment on the Ancient Tree long, long ago...the
enchantment that you have brought upon yourselves because of your
greed. And he who cast the enchantment was a great wizard king, one
who knew the secret of numbers that you so eagerly seek to discover."

The gnomes didn't know what to say. Who was this King Equals
and what did he want from them? Finally the red gnome replied,

"We know the enchantment. Old Dismas explained it to us. And
we know we must count until we discover the secret of numbers. But
these gemstones are far too many for us and when we count we soon
become confused. Can you help us?"

King Equals nodded his head and pulled a magic wand out from beneath his purple cape.

"You're a wizard too?" gasped the green gnome.

"Well, of course. And I can offer you a very simple way to
solve your problem. If you want to find the solution to keeping track
of the numbers you count, you must repeat this verse with me,

Regal old yellowgnomes buy itchy vests.

Simply repeat this verse and the solution will present itself to you."

The gnomes were convinced that King Equals was a fool, but they
soon decided that since they had no better plan there was no harm in
trying the verse, as silly as it seemed. Together they began repeating
the strange words,

Suddenly a radiant light broke through the tops of the trees and
shone upon the mountain of gemstones. And then there was a most
beautiful sight. The reflection of the gemstones created a brilliant
rainbow and a a prism of color filled the sky. Red, orange, yellow,green, blue, indigo, and violet.

"Use these bands of color to help you keep track of your numbers as they grow. But remember, only nine of each kind can fit in one band" instructed King Equals.

Still awestruck by the exquisite color surrounding them, the
gnomes were confused by the words, "Only nine of each kind can fit in
one band." And anticipating their confusion, the King began to
demonstrate.

"Do you see this first band of red? We call this band Units. Each gemstone represents one unit. And we can fill the red band with nine of them."

King Equals picked up a basket and filled it with 9 sparkling gemstones and carefully laid them in the red band of color.

"But there are many more than nine gemstones, Your Majesty. How
can this possibly help us? We have a mountain of gemstones to count"
remarked the confused green gnome.

"Ah, but don't you see these other bands? There are many places within which you may count. When we fill up the red Units we move on to the orange band. This band is called Tens."

"Oh! I know what ten is. We counted ten and made up a symbol
for it in the shape of an X using branches from...Oh. Sorry Dismas.
I'm sure it was not pleasant to have your branches broken off" admitted
the yellow gnome with great regret.

The Ancient Tree lowered his heavy head and offered a gentle smile.

King Equals continued his explanation, "After 9 gemstones fill the Units band we must continue by grouping them together in families of ten."

As he said this, the King carefully lined up the gemstones in groups of ten and placed them in the orange band of color.

"Now remember, only nine of each kind can fit in one band. So once we have nine groups of ten in the Tens place, we must move on to the yellow band of color. We call this the Hundreds."

Beginning to understand the sorting game, the gnomes began
gathering gemstones once again in their baskets. When they filled the Hundredswith nine groups of one hundred gemstones, they moved on to the green. King Equals went on to explain that the green band was called Thousands and the numbers continued far past that. There were blue Ten Thousands, indigo Hundred Thousands and a place in the violet band for a number called Millions.
The gnomes continued their work, and stopped only to notice that King
Equals had opened his purple cape to reveal a funny little symbol made
up of two lines on his chest.

"Excuse me, Your Majesty, but
what is that symbol you wear on your chest? Is it some sort of royal
family crest?" asked the red gnome.

"Oh, it is much more than that. Much more. But that story if
for another time. For now, continue your counting and fill this
rainbow with gemstones. Soon you will learn the purpose of the symbol
I wear upon my chest" answered the King.

The gnomes whispered among themselves, "Well, I sure hope that
mysterious symbol can help us discover the secret of numbers!"

Of course, you know that Ancient Trees can hear the softest,
most faintest sound in the forest. And hearing the musings of the
silly gnomes, Old Dismas let out a laugh that shook the forest floor.

LESSON PLANS

Supplies

There is no need to move on to this lesson until the concepts
introduced in the previous two lessons have been mastered. Move at
your child's pace. The Gnomes and Gnumbers lessons are archived in chronological order on this page for your convenience.

You will need to assemble a Rainbow Tray for this lesson, as well as collect a set of numerals. If you already have a set of moveable numerals
like the ones shown in the picture above, use them. If not, and
purchasing a set is not in your budget, spend one afternoon baking a
set with your child. The easy salt dough recipe is here and a set of cutters
can be purchased. Another option is to model the numerals with modeling
beeswax (Suppliers are linked on the right sidebar.) or Sculpey Clay.

If you already own a Base Ten set,
you can make use of it for place value work. If not, you can download
the Ten Bar and Hundred Flat files below. These were made using Do-A-Dot markers on graph paper. Donna Young's site offers graph paper files for download if your child would like to make a set of base ten gemstones. The graph paper used here is the 10 x 14 size.

Downloads

You can download the story and artwork by clicking on the links below.

We have chosen to present the rainbow in its true order of colors
rather than the simplified version used in many children's songs and
books. Red, Orange, Yellow, Green, Blue, Indigo, and Violet. Have fun
with the mnemonic device created by Marybeth Foss, Regal old yellowgnomes buy itchy vests, for memorizing the colors of the rainbow and their order.

After reading the story and presenting this lesson on place value,
have your child use the gemstones to practice identifying the place
value of each numeral in a larger number. Depending on the age and
ability of the child, you will want to go as high as he or she is
able. A younger child may only understand up to the hundreds place,
while an older child will enjoy discovering a million.

Ask your child to illustrate a place value page in the Main Lesson
Book. The sample rainbow page offered above can be used as a model for
your child.

Place Value Activities

There are three place value activities that you can play this week
using the Rainbow Tray and gemstones. First start by filling a small
basket with gemstones. For instance, if your basket is filled with 23
gemstones, your child will count nine gemstones and, reminding him that
only nine of each kind can be placed in each band, will then cross over
to the Tens color band. The remaining 20 should be grouped in two sets
of ten gemstones. Then show your child how to swap out 10 gemstones for
a ten bar. Continue in this way until your child understands the
concept.

Next, make a 2 or 3-digit number (or larger for an older child)
using the moveable numerals. For instance, arrange the numerals so that
they show 45. Ask you child to build this number with the gemstone
materials or a Base 10 set next to the Rainbow Tray and then place the
numerals in the correct strips of color on the tray.

Another
fun activity is to sort gemstones in different sizes of cups. Begin
with 10 small bathroom-sized cups and ask your child to fill each with
10 gemstones. When ten small cups are filled, they can be poured into
a larger 8 ounce cup, representing 100 (pictured here). Ten of these
8-ounce cups can then be poured into a large 32-ounce cup to represent
1,000. This tactile experience gives the child the opportunity to see
and feel these numbers. The goal is to bring the abstract into the
realm of the concrete.

Fun and Movement

Add movement by playing games of rainbow bean bag toss. Say "Red"
and toss your child a bean bag. He should respond by saying, "Orange"
and then toss the bag back to you. Continue in order to memorize the
sequence of the colors in the rainbow.

Review

Don't forget to review the concepts covered in the previous lessons.
Together read your child's entries in the Main Lesson Book and discuss
each page. Practice Roman Numerals at least one day this week and use
the bean bags to play a game of toss, using odd and even counting as
the rhythm.

If you feel that your child would benefit from math sheet review, there is a wonderful math worksheet generator online called The Math Worksheet Site.
You can get a subscription for only $2.50 per month. (The price is
reduced depending on the length of your subscription.) The
subscription gives you unlimited use of the site and allows you to
create custom math worksheets for your child. The site is very easy to
use. Consider creating sheets on Roman Numerals, odd and even counting,
and place value for your child.

Number Study

This week we will continue our study of numbers by concentrating on
the Number Two. Ask your child to consider the world around him and
find thing that exist in pairs of two. Possible responses might be two
eyes, two ears, two lights (the sun and moon), two feet, etc...

Gemstone Fun for the Little Ones

Nicholas Foss came up with a fun activity for the little ones. Using rainbow-colored craft sticks,
he sorted the different colored gemstones on each stick and made the
pattern of a rainbow. Since these were not glued down, a young child
can repeat this activity many times.

Form Drawing

This week we are adding the element of form drawing. This exercise
in creating rhythmic patterns prepares a young child for handwriting
and improves handwriting for the older child. When presenting these
running forms, allow your child to first practice in a shallow tray of
sand or cornmeal. Tracing the form with a finger in the air also helps
the child in learning to imitate the pattern. Your child should use
stick crayons when drawing the forms. Later in this main lesson we
will be painting the geometric shapes and practice in form drawing will
help in preparing the child for this activity.

One
running form will be introduced each week for a period of nine weeks
and we will be using the art of story telling to capture the attention
and interest. Read the following story to your child as an
introduction to the first running form. If your child has not had much
practice in drawing, it may be a good idea to spend some time
practicing standing forms such as squares, triangles, circles,
rectangles and stars. Draw with your child and allow her to imitate
you. Don't hesitate to draw a form for a young child, allowing her to
trace your work. Soon they will learn the movement and pattern and be
able to work independently.

Our form drawing tale will be centered on the story of King Equals
and how he brought peace to the Kingdom of Numeria. Read this story to
your child and practice drawing the form before making a main lesson
book page (a sample page is shown here). Perhaps after practicing the
form each day, the child can draw it in the Main Lesson Book on Friday.
Encourage your child to draw this form with a continuous and
uninterrupted movement. Have fun with the colors. You can use these
forms to practice memorizing the order of the colors of the rainbow too.

The Story of the Great King Equals

Once upon a time there was a sad prince who lived in a great stone
castle in a divided kingdom. For many years his ancestors ruled the
kingdom in peace; but one day, in the furthest part of the kingdom, a
great treasure was discovered in hidden cave. Ever since that day the
people of the kingdom argued and quarreled over the question of who was
the rightful owner of the cave and the great treasure. As the prince
looked out through the parapets of the great stone castle, he longed
for the day when his family's great kingdom would be one again.

Having learned that with the help of a beautiful rainbow
it was easy to recognize even the largest of numbers, the four gnomes
went back to their work of trying to unlock the secret of numbers.

"We now know that some numbers are odd, while others are even,"
recalled the yellow gnome. "And we also know that no matter how large a
number is, we can figure it out with the help of the bands of the
rainbow. But the enchantment has not yet been broken. We obviously
haven't figured out the secret of numbers. What else could there be?"

As soon as the yellow gnome finished speaking, the green gnome
walked to the base of the mountain of gemstones, picked up the largest
basket he could find and filled it to the top. Then taking the
gemstones out, one by one, he lined them on the ground and began,

"One gemstone and one gemstone and one gemstone..." he continued for quite some time.

Curious and confused, the other three gnomes sat down and
watched their green friend continue. Soon the silence was broken by
the woodsy "Huuuuumph!" of Old Dismas.

"Addition."

The green gnome didn't hear the Ancient Tree and continued, "...and one gemstone and one gemstone."

Old Dismas spoke up, "Did you know that what you are doing has a name?"

Suddenly the green gnome stopped. He had been certain that what
he was doing was something very new and had been discovered by him
alone.

"Addition. What you are doing is called Addition" repeated Old Dismas. "You see how you are adding the gemstones, one gemstone and one gemstone and so on and so forth? That is called Addition. How many gemstones do you now have?" asked Old Dismas.

"Well, let's see. I...I don't know. One gemstone and one
gemstone and one gemstone..." The green gnome realized that while he
could continue to add one gemstone after another to his long line, he
was no closer to knowing how many he had counted.

Walking over to the green gnome, King Equals once again reached inside his cape and pulled out his magic wand.

"I have a gift for you that will help you to answer your
question." The purple king waved his wand and pointed it towards the
chest of the little green gnome. Suddenly a strange symbol appeared.
It was made up of two lines, one going up and down and the other going
side to side. And while the little lines went in opposite directions,
they crossed exactly in the middle.

"Whenever you want to find out how many, just remember to use
this sign between each number. And when you have finished, call on me
and you will find your answer" finished King Equals.

The green gnome looked again at his long line of gemstones and this
time he traced his new sign in the earth between each gemstone. "One
gemstone + one gemstone. King Equals?"

The king approached the gnome and answered, "Two gemstones. One gemstone and one gemstone equals two gemstones."

The green gnome continued, "Two gemstones + one gemstone. Equals?"

King Equals answered, "Two gemstones and one more gemstone is the same as three gemstones."

The little green gnome soon understood that the funny little
sign joined the gemstones together and made them one. Not only that,
but King Equals' answer told him the number that they made.

"Oh! I get it. And three gemstones and one gemstone is four
gemstones!" The green gnome was so excited. He merrily danced around
his work.

"Very good, my friend. You have understood well. And because
you have figured out this mystery of counting, I give you a new name.
From this time forth you will be known in all Numeria as Plus." King Equals tapped the shoulder of of the happy gnome, bestowing upon him his new name.

"I guess I've figured it out then. Addition is the secret of numbers" announced Plus, rising from a ridiculously low bow.

"Not so fast," replied Old Dismas, chuckling to himself. "You
are one step closer, that is true; but there is still much more to
learn. And until then, I suggest that you and your friends practice
using your new symbol, for it will help you in unlocking this great
secret."

The red and yellow gnomes rushed to Plus' side, ready to play
the new counting game. But not the blue gnome. No, he wasn't
interested in the game of Plus. Instead, the blue gnome quietly
tip-toed around to the back of the mountain of gemstones.

"I have a better way" he laughed to himself as he began filling the pockets of his cape with handfuls of colorful gemstones.

LESSON PLANS

So far we've learned about Roman Numerals, odd and even numbers, place value
and the process of addition. At this point the story of the four greedy
gnomes stops for a few weeks in order to allow the children an
opportunity to absorb the material. During the next three weeks new
ideas on teaching the concept of addition and extension activities will
be posted, along with the next three installment of the Form Drawing
stories and the next three assignments in our Number Study.

Each week we will go more
in-depth with the concept of addition, but before we continue in this
study it is important to make sure that your child understands the
purpose and function of adding and making sums. Explain that the
addition symbol shows that we are joining the two things together that
are on either side of it and that the symbol for equals simply means
"is the same as." Ask your child to make a page in his Main Lesson
Book modeled after the one shown above. Concentrate this week on
adding 0 and 1 to a number. Depending on the age and ability of your
child, you can also make a +1 page in the Main Lesson Book (0+1 through
11+1 is enough). Allow your child to use the gemstones for practice.
Fill one basket with a small amount of gems and a second basket with
one. Ask your child to count the gems together to find the answer.
You can illustrate these problems on the blackboard or let your child
use Do-A-Dots to illustrate addition with gemstones on paper.

Add rhythm and movement by
playing addition bean bag toss. Say, "One plus one is..." and toss the
bean bag to your child. The child should respond with the sum, "Two!"
and then say, "Two plus one is..." and toss the bean bag back to you.
Continue in this way as long as your child is enjoying the game.
(Remember to play games of bean bag toss to practice counting forward
and backward from 0-100, odd and even counting, and the colors of the
rainbow.)

This week we continue our lessons in Form Drawing by telling the second part of The Story of the Great King Equals. Tell
this story to your child as many times as necessary until he can
successfully draw the new running form. By the end of the week, add
this form to the Main Lesson Book. Keep in mind that these forms
should be drawn with stick crayons in one continuous stroke. You can
download the story with picture here (Download form_drawing_2.pdf

). First review last week's story and then continue with the following:

Looking through the parapets of the great stone castle, the
prince saw in the distance an immense range of mountains. "I will
depart from this divided kingdom and travel to the mountains, in search
of a wise man who can tell me how to unite my family's kingdom once
again" decided the purple prince. And with much hope, the prince left
his home in the great stone castle and set off on his horse over the misty mountains.

The running form in the drawing this week is similar in appearance
to a mountain range. Point this out to your child and then practice
drawing the form with him. Encourage your child to use these forms when
making page borders in their Main Lesson Book.

NUMBER STUDY

This week we will also continue
our study of numbers by considering the number three. Ask your child
to think of things that are three. St. Patrick's 3-leaf clover that
signifies the Three Persons of the Holy Trinity, and a triangle are
three examples. Have your child make a number three page for her Main
Lesson Book.

Continue spending time reading math-themed living books. Last week
we suggested a few titles. If you have not yet read them with your
child, go back to that list and read through. Now is a good time to
review the things we've learned in the past four weeks and reading the
concepts in a living picture book will help reinforce the concepts.

Each week during our four-week addition unit, more theme-related books will be suggested. The following books are this week's suggestions for living books on the topic of addition. Try to read one each day.

The exciting mathematical tale of Gnomes and Gnumbers
will continue in January 2008, but that doesn't mean our work is done.
Far from it! From now until then we have much work to do with our new
friend Plus. Our clever green friend has made an important discovery
and for the following four weeks, assuming that you will be breaking
for the weeks of both Thanksgiving and Christmas, we offer the
following extensions to further the study of the process of Addition.

For the child who is just beginning her study of
addition, you will want to spend the next month working with simple
addition equations and memorizing the addition tables. Begin with the
+1 table and continue up until +9. The use of manipulatives, whether
the gnomes' gemstones or a Base Ten set, is an essential element in
presenting an abstract concept in a concrete way.

Older children who already have a solid understanding will benefit from the Whole to Parts exercises listed below and can increase their skills by working on Addition with Regrouping topics using a Base Ten set. If you do not already own a Base Ten set, a gemstone base ten set for download was offered in Lesson Three of Gnomes and Gnumbers. If you have not already, you may want to download those files and prepare them.

Along with games of bean bag toss, review odd and even numbers by filling baskets with different amounts of gemstones
and asking your child to sort the gems in groups of twos. Remind her
that if one is left over, the number is odd. An alternative review
activity is to write numbers on paper or the blackboard and ask your
child to circle all odd numbers in one color and all even numbers in a
different color. When analyzing larger numbers, be sure to remind
your child that the numeral in the units place is the only numeral that
we consider when determining whether or not a number is odd or even.

Continue to review the concept of place value by
using the rainbow tray. Make different numbers by placing the numerals
within the bands of the rainbow. Ask your child to identify the place
value of the numerals. For instance, if you demonstrate the number 1,
325 on the rainbow tray, ask your child questions such as "Which number
is in the hundred's place?" or "In what place is the three?" You can
extend the questions to include, "If a two is in the ten's place, what
is the value of the two?" The answer is two tens or twenty. Continue
the activity from Lesson Three of building numbers with the Base Ten
set or gemstones and asking your child to place the correct numerals on
the rainbow tray. You can reverse this activity by making a number on
the tray and asking your child to build it with the manipulatives.
(Review Lesson Three for more explanation on place value activities.)

ADDITION EXTENSIONS

Addition Strip Board

The Addition Strip Board, a great aid in comprehending and memorizing
the addition tables, is a traditional Montessori resource. You can
purchase a wooden board and strips for a discounted price at Alison's Montessori or you can download a free PDF from Montessori Materials.
(Click on "Addition Strip Board" under the heading "Addition
Material.") For maximum durability, we suggest printing on card stock
and mounting the print-outs on foam board. As beautiful as the
traditional set is, this more economical paper alternative is just as
effective as using the more expensive wooden set. You will also want to
print out this free set of Addition Equation Slips (Click on "Addition Combination Slips.") or purchase the matching wooden set from Alison's.

To introduce the board to a child, ask him to randomly choose an
addition equation slip. In the picture here, the equation "4+3=" was
chosen. The equation was built by lining the blue 4 strip on the first
line, followed by the red 3 strip. With both strips lined up together,
it is obvious by checking the number line that the sum is 7. The
number seven tile was placed after the equation and the sum was
recorded on graph paper.

Using graph paper for recording equations helps the child line
up the numbers in the equation. (It is even more helpful for advanced
topics such as multiple-digit addition or long division.) For free graph paper
you can visit Donna Young's site. There you will find many different
sizes of graph paper for download. For a young child, the larger
squares are best. If you would prefer a bound main lesson book filled with graph paper you can find one at Paper, Scissors, Stone. (Scroll down to find the "Arithmetic Book.")

Spend the next few weeks introducing and memorizing the addition tables
+1 through +9. You can use the equation slips to introduce each table,
as well as for daily practice. For instance, if your child is going to
be working on the +4 table, collect the equation slips that express the
+4 equations. Ask your child to build the equations on the strip
board. For further practice, the equations slips can be used as
flashcards as your child works on memorizing the addition tables. And
if he can't remember the sum, always ask him to build the equation on
the board.

Don't forget to add fun and movement by playing
games of addition bean bag toss. Using a small bean bag, toss the bag
to your child as you say "One plus one equals..." Your child should
respond with the sum, "Two," as he tosses it back to you. Continue up
to 1+9. Use this game as a memory aid for all 9 addition tables.

Addition with Regrouping

Using a Base Ten set, you can teach your older child how to add
larger numbers that require regrouping or carrying. Consider printing
multiple-digit addition equation sheets from The Math Worksheet Site
or write your own on graph paper. Use the Base Ten materials to
demonstrate the process, reminding your child of the counsel of King
Equals, "Only nine of each kind can fit in one band."

Begin with a problem that requires regrouping such as "35+68=" and
build each number with the gemstone materials or your Base Ten set.

Pointing
out that we only add together those of like kind and that we always
start with the Units, combine the Units into one pile.

Counting the Units we find that there are 13 cubes. Remind your
child of the wise counsel of King Equals, "Only nine of each kind can
fit in one band." Thirteen is more than nine and cannot fit in the red
units band of our place value rainbow. We're going to have to do some
trading. Get out a Ten Bar and line up ten of the thirteen units
against it. Trade then Ten Bar for the ten Units.

Remind your child that we can only add those of like kind, therefore
the Ten Bar must be added to the pile of Tens and not the Units. We
now only have three Units and move on to adding the Tens together.

"Only nine of each kind can fit in one band." Since we have ten
we'll have to trade again. Get out a Hundred Flat and trade the ten Ten
Bars for one Hundred.

We've stayed true to King Equal's instruction and now have three
Units and one Hundred. Ask your child to use the numerals and place
them in the correct bands of the rainbow. (You can bake these numerals, buy a set of moveable numerals or write them on the blackboard or paper in the different colors of each band.)

This example is one in which there is a zero in the Ten's
place. It is important to point out that when we trade the different
values and end up with nothing in one of the places, we need to make
sure that we use a zero to hold the place. Demonstrate with this
example that if you didn't use the zero to hold the Ten's Place, the
number would no longer be one hundred and three, but thirteen. The
child can clearly see that more than 13 is represented by the blocks.

When recording these problems we recommend using graph paper.
When carrying numbers above the equation, it is much easier to keep
track of the Tens or Hundreds that we've carried over if we can line
them up in the square above the Ten's or Hundred's Place. (Sources for
graph paper are listed above.)

Whole to Parts

The following exercise is recommended for all children, even those who have experience working with addition equations.

Rather than simply memorize addition equations and their sums, it
is important to study numbers in a holistic way by considering them from the whole to the parts.
What is the number 12? It can be many things, such as 6+6 or 9+3; but
6+6 can only ever be 12, just as 9+3 can only ever be 12. The focus in
this study is to consider what makes up the WHOLE number and for this
reason, even a child who has memorized the addition tables can benefit
from this exercise. An understanding of the whole of numbers serves in building a stronger foundation for more advanced work in the future.

Over the course of the next month spend a separate day on each of
the numbers, 1 through 12. Consider each number as a whole and from
that starting point, pick out its parts. There are two ways to have
fun with this exercise. The first is to draw Number Ladders.

By drawing a simple ladder we can imagine that the number we are
considering is walking up its rungs. Each rung represents a different
way to make up that number. Twelve can be eleven and one or it can be
five and seven. The ladder shows the many parts of the number. Point
out the patterns that numbers make. Going down the left side of the
ladder we see that the numbers increase from zero to twelve, while on
the right side the numbers decrease from twelve to zero.

Ask you child to make a number ladder for the numbers one to
twelve. When he has finished this over a series of many days, consider
the twelve number ladders together and compare them. For instance, the
eight ladder has many more rungs than the two ladder. Pointing out
these patterns and differences transforms an abstract concept into a
concrete reality.

A book such as 12 Ways to Get 11 is a great example of expressing the parts of numbers in pictoral form. Make a number picture
for each of the twelve numbers. Perhaps seven is a scene of a house
with one door, two windows, two trees, one sun, and one chimney.
(1+2+2+1+1=7) If your child enjoys this activity, consider using the
blackboard for repeated play or maybe he'll want to make his own number
picture book.

Number Study: Four

This week we will continue our
study of numbers by considering the number four. Ask your child to
think of things that are four. The four inspired authors of the
Gospels, Saints Matthew, Mark, Luke and John; the four cardinal
directions, north, south, east and west; and the four sides of a square
or rectangle are all possible examples. Have your child make a Number
Four page for his Main
Lesson Book.

Making Numbers

Shaping the numerals out of modeling clay
is a beneficial tactile experience for all learners, but especially for
those with learning challenges such as dyslexia. Later this week
Elizabeth will be posting about modeling the numbers as well as
modeling representations of number sentences.

When considering an equation, such as four plus two, it is helpful
for a child to model the numeral four, an addition symbol, the numeral
two and an equals symbol and answer the question by rolling out small
balls to demonstrate the equation. Four small balls grouped with two
small clay balls adds up to six. The equation is visually represented
and the child has been offered the experience of forming the numerals
with her own hands. Just as the exercise of Number Study gives the
child an opportunity to contemplate the reality of the number in the
created world, so too the exercise of making numbers offers the actual
experience of forming the individual numbers and feeling its shape.

Encourage an older child to participate in this activity by
pairing him up with a younger sibling. In teaching and directing a
younger child, the older child not only increases his understanding,
but demonstrates it.

Form Drawing

Continue the lessons in Form Drawing by reading the next installment
in the tale "The Story of the Great King Equals." Who is King Equals?
Where did he come from? And how did he get that funny symbol? The
answers to these questions and more will be discovered in this tale.
After reading this week's portion of the story, draw with your child
and allow her to imitate you. Don't hesitate to
draw a form for a young child, allowing her to trace your work. Soon
she will learn the movement and pattern and be able to work
independently. Encourage your child to draw this form using a stick
crayon with a continuous and
uninterrupted motion.

Descending toward the base of the misty mountains, the Purple Prince
looked out upon the vast land. As far as the eye could see were
rolling hills covering the land. Spurring his loyal horse to continue
on their journey, with great difficulty the purple prince rode over
large hills. He was greatly relieved to discover a stretch of small
hills. At last he came to a series of medium-sized hills. Finishing
this leg of the journey and dismounting his horse, the purple prince
was dismayed at what lay before him. Would he ever find a wise man to
help him unite his kingdom?